Question: Subtract the following rational expressions. $\dfrac{4y+3}{-2y+5}-\dfrac{7}{9y^4}=$
Solution: We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({-2y+5})\cdot({9y^4})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{4y+3}{{-2y+5}}-\dfrac{7}{{9y^4}} \\\\ &=\dfrac{(4y+3)\cdot({9y^4})}{({-2y+5})\cdot({9y^4})}-\dfrac{7\cdot({-2y+5})}{({9y^4})\cdot({-2y+5})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{(4y+3)\cdot(9y^4)}{(-2y+5)\cdot(9y^4)}-\dfrac{7\cdot(-2y+5)}{(9y^4)\cdot(-2y+5)} \\\\ &=\dfrac{(4y+3)\cdot(9y^4)-7\cdot(-2y+5)}{(-2y+5)(9y^4)} \\\\ &=\dfrac{36y^5+27y^4+14y-35}{(-2y+5)(9y^4)} \end{aligned}$ In conclusion, $\dfrac{4y+3}{-2y+5}-\dfrac{7}{9y^4}=\dfrac{36y^5+27y^4+14y-35}{(-2y+5)(9y^4)}$